Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(n)
Real (Kind=nag_wp), Intent (In)	::	a(n), x(n), tol
Real (Kind=nag_wp), Intent (Out)	::	p(n), q(n)

C Header Interface

#include <nag.h>

void	s14bnf_ (const Integer n, const double a[], const double x[], const double tol, double p[], double q[], Integer ivalid[], Integer *ifail)

The routine may be called by the names s14bnf or nagf_specfun_gamma_incomplete_vector.

3 Description

s14bnf evaluates the incomplete gamma functions in the normalized form, for an array of arguments

a_{i}, x_{i}

, for

i = 1, 2, \dots, n

P (a, x) = \frac{1}{Γ (a)} \int_{0}^{x} t^{a - 1} e^{- t} d t,

Q (a, x) = \frac{1}{Γ (a)} \int_{x}^{\infty} t^{a - 1} e^{- t} d t,

with

x \geq 0

and

a > 0

, to a user-specified accuracy. With this normalization,

P (a, x) + Q (a, x) = 1

Several methods are used to evaluate the functions depending on the arguments

a

and

x

, the methods including Taylor expansion for

P (a, x)

, Legendre's continued fraction for

Q (a, x)

, and power series for

Q (a, x)

. When both

a

and

x

are large, and

a ≃ x

, the uniform asymptotic expansion of Temme (1987) is employed for greater efficiency – specifically, this expansion is used when

a \geq 20

and

0.7 a \leq x \leq 1.4 a

Once either

P

Q

is computed, the other is obtained by subtraction from

1

. In order to avoid loss of relative precision in this subtraction, the smaller of

P

and

Q

is computed first.

This routine is derived from the subroutine GAM in Gautschi (1979b).

4 References

Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software 5 466–481

Gautschi W (1979b) Algorithm 542: Incomplete gamma functions ACM Trans. Math. Software 5 482–489

Temme N M (1987) On the computation of the incomplete gamma functions for large values of the parameters Algorithms for Approximation (eds J C Mason and M G Cox) Oxford University Press

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of points.

Constraint:

n \geq 0

2: $a (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

a_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

a (i) > 0

, for

i = 1, 2, \dots, n

3: $x (n)$ – Real (Kind=nag_wp) array Input

On entry: the argument

x_{i}

of the function, for

i = 1, 2, \dots, n

Constraint:

x (i) \geq 0

, for

i = 1, 2, \dots, n

4: $tol$ – Real (Kind=nag_wp) Input

On entry: the relative accuracy required by you in the results. If s14bnf is entered with tol greater than

1.0

or less than machine precision, then the value of machine precision is used instead.

5: $p (n)$ – Real (Kind=nag_wp) array Output

On exit:

P (a_{i}, x_{i})

, the function values.

6: $q (n)$ – Real (Kind=nag_wp) array Output

On exit:

Q (a_{i}, x_{i})

, the function values.

7: $ivalid (n)$ – Integer array Output

On exit:

ivalid (i)

contains the error code for

a_{i}

and

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $a_{i} \leq 0$ .
$ivalid (i) = 2$: $x_{i} < 0$ .
$ivalid (i) = 3$: Algorithm fails to terminate.

8: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, at least one value of x was invalid.
Check ivalid for more information.

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

There are rare occasions when the relative accuracy attained is somewhat less than that specified by argument tol. However, the error should never exceed more than one or two decimal places. Note also that there is a limit of

18

decimal places on the achievable accuracy, because constants in the routine are given to this precision.

8 Parallelism and Performance

s14bnf is not threaded in any implementation.

9 Further Comments

The time taken for a call of s14bnf depends on the precision requested through tol, and n.

10 Example

This example reads values of a and x from a file, evaluates the functions at each value of

a_{i}

and

x_{i}

and prints the results.

s14bn: FL CL CPP AD

NAG FL Interfaces14bnf (gamma_​incomplete_​vector)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s14bnf (gamma_incomplete_vector)